The last decade saw an increasing interest in the problems of soft condensed matter, also know as complex fluids. A particular important class is charged polymers, called polyelectrolytes that dissociate charges in aqueous solution. In the beginning we will introduce these materials, and present several important examples occurring in daily life. Due to their complex interactions, computer simulations have proven to be an indispensable tool for understanding the properties of these materials. We introduce the underlying models, present the questions which can be asked and answered, and demonstrate some recent astonishing findings which show that under certain circumstances like charged bodies can attract each other, and charged bodies can be overcharged. By means of computer generated video animations of simulations one gains valuable insight into the complex behavior of charged macromolecules. We will conclude with a short overview of other interesting phenomena occurring in the nano-world of charged soft matter systems.